## Engage NY Eureka Math Algebra 2 Module 3 Lesson 8 Answer Key

### Eureka Math Algebra 2 Module 3 Lesson 8 Opening Exercise Answer Key

Opening Exercise:

a. Evaluate each expression. The first two have been completed for you.

i. WhatPower_{2}(8) = 3

Answer:

3, because 2^{3} = 8

ii. WhatPower_{3}(9) = 2

Answer:

2, because 3^{2} = 9

iii. WhatPower_{6}(36) = ______

Answer:

2, because 6^{2} = 36

iv. WhatPower_{2}(32) = _____

Answer:

5, because 2^{5} = 32

v. WhatPower_{10}(1000) = _____

Answer:

3, because 10^{3} = 1000

vi. WhatPower_{10}(1000000) = _____

Answer:

6, because 10^{6} = 1,000,000

vii. WhatPower_{100}(1000000) = _____

Answer:

3, because 100^{3} = 1,000,000

viii. WhatPower_{4}(64) = _____

Answer:

3, because 4^{3} = 64

ix. WhatPower_{2}(64) = _____

Answer:

6, because 2^{6} = 64

x. WhatPower_{9}(3) = _____

Answer:

\(\frac{1}{2}\) because 9^{\(\frac{1}{2}\)} = 3

xi. WhatPower_{5}(√5) =_____

Answer:

\(\frac{1}{2}\) because 5^{\(\frac{1}{2}\)} = √5

xii. WhatPower_{\(\frac{1}{2}\)}(\(\frac{1}{8}\)) = ___

Answer:

3, because \(\left(\frac{1}{2}\right)^{3}\) = \(\frac{1}{2}\)

xiii. WhatPower_{42}(1) = ____

Answer:

0, because 42^{0} = 1

xiv. WhatPower_{100}(0.01) = ___

Answer:

-1, because 100^{-1} = 0.01

xv. WhatPower_{2}(\(\frac{1}{4}\))

Answer:

-2, because 2^{-2} = \(\frac{1}{4}\)

xvi. WhatPower_{\(\frac{1}{4}\)}(2) = _____

Answer:

\(\frac{1}{4}\) because (\(\frac{1}{4}\))^{\(-\frac{1}{2}\)} = 4^{\(\frac{1}{2}\)} = 2

b. With your group members, write a definition for the function WhatPowerb, where b is a number.

Answer:

The value of WhatPower_{b} is the number you need to raise b to in order to get x. That is, if b^{L} = x, then L = WhatPower_{b}(x).

### Eureka Math Algebra 2 Module 3 Lesson 8 Exercise Answer Key

Exercises 1 – 9:

Evaluate the following expressions, and justify your answers.

Exercise 1.

WhatPower_{7}(49)

Answer:

WhatPower_{7}(49) = 2 because 7^{2} = 49

Exercise 2.

WhatPower_{0}(7)

Answer:

WhatPower_{0}(7) does not make sense because there is no power of 0 that will produce 7.

Exercise 3.

WhatPower_{5}(1)

Answer:

WhatPower_{5}(1) = 0 because 5^{0} = 1.

Exercise 4.

WhatPower_{1}(5)

Answer:

WhatPower_{1} (5) does not exist because for any exponent L, 1^{L} = 1, so there is no power of 1 that will produce 5.

Exercise 5.

WhatPower_{-2}(16)

Answer:

WhatPower_{-2}(16) = 4 because (-2)^{4} = 16.

Exercise 6.

WhatPower_{-2}(32)

Answer:

WhatPower_{-2}(32) does not make sense because there is no power of -2 that will produce 32.

Exercise 7.

WhatPower_{\(\frac{1}{3}\)}(9)

Answer:

WhatPower_{\(\frac{1}{3}\)}(9) = -2 because (\(\frac{1}{3}\))^{-2} = 9

Exercise 8.

WhatPower_{\(-\frac{1}{3}\)}(27)

Answer:

WhatPower_{\(-\frac{1}{3}\)}(27) does not make sense because there is no power of –\(\frac{1}{3}\) that will produce 27.

Exercise 9.

Describe the allowable values of b in the expression WhatPowerb(x). When can we define a function f(x) = WhatPower_{b}(x)? Explain how you know.

Answer:

1f b = 0 or b = 1, then the expression WhatPowerb(x) does not make sense. If b < 0, then the expression WhatPower_{b}(x) makes sense for some values of x but not for others, so we cannot define a function f(x) = WhatPower_{b}(x) if b < 0. Thus, we can define the function f(x) = WhatPower_{b}(x) if b > 0 and b ≠ 1.

Examples:

Example 1.

log_{2}(8) = 3

Answer:

3, because 2^{3} = 8

Example 2.

log_{3}(9) = 2

Answer:

2, because 3^{2} = 9

Example 3.

log_{6}(36) = ___

Answer:

2, because 6^{2} = 36

Example 4.

log_{2}(32) = ___

Answer:

5, because 2^{5} = 32

Example 5.

log_{10}(1000) = ___

Answer:

3, because 10^{3} = 1000

Example 6.

log_{42}(1) = ___

Answer:

0, because 42^{0} = 1

Example 7.

log_{100}(0.01) = ___

Answer:

-1, because 100^{-1} = 0.01

Example 8.

log_{2}(\(\frac{1}{4}\)) = ___

Answer:

-2, because 2^{-2} = \(\frac{1}{4}\)

Exercise 10.

10.

Compute the value of each logarithm. Verify your answers using an exponential statement.

a. log_{2}(32) = ___

Answer:

log_{2}(32) = 5, because 2^{5} = 32

b. log_{3}(81) = ___

Answer:

log_{3}(81) = 4, because 3^{4} = 81

c. log_{9}(81) = ___

Answer:

log_{9}(81) = 2, because 9^{2} = 81

d. log_{5}(625) = ___

Answer:

log_{5}(625) = 4, because 5^{4} = 625

e. log_{10}(1,000,000,000) = ___

Answer:

log_{10}(1,000,000,000) = 8, because 10^{9} = 1,000,000,000.

f. log_{1000}(1000000000) = ___

Answer:

log_{1000}(1000000000) = 3, because 1000^{3} = 10000000000

g. log_{13}(13) = ___

Answer:

log_{13}(13) = 1, because 13^{1} = 13

h. log_{13}(1) = ___

Answer:

log_{13}(1) = 0, because 13^{0} = 1.

i. log_{7}(√7) = ___

Answer:

log_{7}(√7) = 0, because 7^{\(\frac{1}{2}\)} = √7.

j. log_{9}(27) = ___

Answer:

log_{9}(27) = \(\frac{3}{2}\), because 9^{\(\frac{3}{2}\),} = 3^{3} = 27.

k. log_{√7}(7) = ___

Answer:

log_{√7}(7) = 2, because (√7)^{2} = 7.

l. log_{√7}(\(\frac{1}{49}\)) = ___

Answer:

log_{√7}(\(\frac{1}{49}\)) = -4, because √7^{-4} = \(\frac{1}{(\sqrt{7})^{4}}=\frac{1}{49}\).

m. log_{x}(x^{2}) = ___

Answer:

log_{x}(x^{2}) = 2, because (x)^{2} = x^{2}.

### Eureka Math Algebra 2 Module 3 Lesson 8 Problem Set Answer Key

Question 1.

Rewrite each of the following in the form WhatPower_{b}(x) = L.

a. 3^{5} = 243

Answer:

WhatPower_{3}(243) = 5

b. 6^{-3} = \(\frac{1}{216}\)

Answer:

WhatPower_{6}(\(\frac{1}{216}\)) = -3

c. 9^{0} = 1

Answer:

WhatPower_{9}(1) = 0

Question 2.

Rewrite each of the following in the form log(x) = L.

a. 16^{\(\frac{1}{4}\)} = 2

Answer:

log_{16}(2) = \(\frac{1}{4}\)

b. 10^{3} = 1,000

Answer:

log_{10}(1,000) = 3

c. b^{k} = r

Answer:

log_{b}(r) = k

Question 3.

Rewrite each of the following in the form b^{L} = x.

a. log_{5}(625) = 4

Answer:

5^{4} = 625

b. log_{10}(0.1) = -1

Answer:

10^{-1} = 0.1

c. log_{27}9 = \(\frac{2}{3}\)

Answer:

27^{\(\frac{2}{3}\)} = 9

Question 4.

Consider the logarithms base 2. For each logarithmic expression below, either calculate the value of the expression or explain why the expression does not make sense.

a. log_{2}(1024)

Answer:

10

b. log_{2}(128)

Answer:

7

c. log_{2}(√8)

Answer:

\(\frac{3}{2}\)

d. log_{2}(\(\frac{1}{16}\))

Answer:

-4

e. log_{2} (0)

Answer:

This does not make sense. There is no value of L so that 2^{L} = 0

f. log_{2}(\(-\frac{1}{32}\))

Answer:

This does not make sense. There is no value of L so that 2^{L} is negative.

Question 5.

Consider the logarithms base 3. For each logarithmic expression below, either calculate the value of the expression or explain why the expression does not make sense.

a. log_{3}(243)

Answer:

5

b. log_{3}(27)

Answer:

3

c. log_{3}(1)

Answer:

0

d. log_{3}(\(\frac{1}{3}\))

Answer:

-1

e. log_{3}(0)

Answer:

This does not make sense. There is no value of L so that 3^{L} = 0.

f. log_{3}(\(-\frac{1}{3}\))

Answer:

This does not make sense. There is no value of L so that 3^{L} < 0.

Question 6.

Consider the logarithms base 5. Fo.’ each logarithmic expression below, either calculate the value of the expression or explain why the expression does not make sense.

a. log_{5}(3125)

Answer:

5

b. log_{5}(25)

Answer:

2

c. log_{5}(1)

Answer:

o

d. log_{5}(\(\frac{1}{25}\))

Answer:

-2

e. log_{5}(0)

Answer:

This does not make sense. There is no value of L so that 5^{L} = o

f. log_{5}(-\(\frac{1}{25}\))

Answer:

This does not make sense. There is no value of L so that ^{L} is negative.

Question 7.

Is there any positive number b so that the expression log(0) makes sense? Explain how you know.

Answer:

No, there is no value of L so that b^{L} = 0. I know b has to be a positive number. A positive number raised to an exponent never equals 0.

Question 8.

Is there any positive number b so that the expression log_{b}(-1) makes sense? Explain how you know.

Answer:

No. Since b is positive, there is no value of L so that b^{L} is negative. A positive number raised to an exponent never has a negative value.

Question 9.

Verify each of the following by evaluating the logarithms.

a. log_{2}(8) + log_{2}(4) = log_{2}(32)

Answer:

3 + 2 = 5

b. log_{3}(9) + log_{3}(9) = log_{3}(81)

Answer:

2 + 2 = 4

c. log_{4}(4) + log_{4}(16) = log_{4}(64)

Answer:

1 + 2 = 3

d. log_{10}(10^{3}) + log_{10}(10^{4}) = log_{10}(10^{7})

Answer:

3 + 4 = 7

Question 10.

Looking at the results from Problem 9, do you notice a trend or pattern? Can you make a general statement about the value of log(x) + log(y)?

Answer:

The sum of two logarithms of the same base is found by multiplying the input values,

log_{b}(x) + log_{b}(y) = log_{b}(xy). (Note to teacher: Do not evaluate this answer harshly. This is just a preview of a property that students learn later in the module.)

Question 11.

To evaluate log_{2}(3), Autumn reasoned that since log_{2}(2) = 1 and log_{2}(4) = 2, log_{2}(3) must be the average of 1 and 2 and therefore log2 (3) = 1. 5. Use the definition of logarithm to show that log_{2}(3) cannot be 1. 5. Why is her thinking not valid?

Answer:

According to the definition of logarithm, log_{2}(3) = 1.5 only if 2^{1.5} = 3. According to the calculator, 2^{1.5} ≈ 2.828, so log_{2}(3) cannot be 1.5. Autumn was assuming that the outputs would follow a linear pattern, but since the outputs are exponents, the relationship is not linear.

Question 12.

Find the value of each of the following.

a. If x = log_{2}(8) and y = 2^{x}, find the value of y.

Answer:

y = 8

b. If log_{2}(x) = 6, find the value of x.

Answer:

x = 64

c. If r = 2^{6} and s = log_{2}(r), find the value of s.

Answer:

s = 6

### Eureka Math Algebra 2 Module 3 Lesson 8 Exit Ticket Answer Key

Question 1.

Explain why we need to specify 0 < b < 1 and b > 1 as valid values for the base b in the expression log_{b}(x).

Answer:

If b = 0, then log_{0}(x) = L means that 0^{L} = x, which cannot be true if x ≠ 0.

If b = 1, then log_{1}(x) = L means that 1^{L} = x, which cannot be true if x ≠ 1.

If b < 0, then log_{b}(x) = L makes sense for some but not all values of x > 0; for example, if b = -2 and x = 32, there is no power of -2 that would produce 32, so log_{-2}(32) does not makes sense.

Thus, if b ≤ 0 or b = 1, then for many values of x, the expression log_{b}(x) does not make sense.

Question 2.

Calculate the following logarithms.

a. log_{5}(25)

Answer:

log_{5}(25) = 2

b. log_{10}(\(\frac{1}{100}\))

Answer:

log_{10}(\(\frac{1}{100}\)) = -2

c. log_{9}(3)

Answer:

log_{9}(3) = \(\frac{1}{2}\)